Question

which number line represents the solution set for the inequality $-4(x + 3) leq -2 - 2x$?

Explanation:

Step1: Expand the left - hand side

Expand \(-4(x + 3)\) using the distributive property \(a(b + c)=ab+ac\).
\(-4(x + 3)=-4x-12\). The inequality becomes \(-4x-12\leq - 2-2x\).

Step2: Add \(4x\) to both sides

To get the \(x\) terms on one side, add \(4x\) to both sides of the inequality.
\(-4x + 4x-12\leq-2-2x + 4x\). Simplifying gives \(-12\leq - 2+2x\).

Step3: Add \(2\) to both sides

Add \(2\) to both sides to isolate the term with \(x\).
\(-12 + 2\leq-2 + 2+2x\). So, \(-10\leq2x\).

Step4: Divide both sides by \(2\)

Divide both sides of the inequality \(-10\leq2x\) by \(2\).
\(\frac{-10}{2}\leq\frac{2x}{2}\). This simplifies to \(-5\leq x\) or \(x\geq - 5\).

Answer:

The number line with a closed - circle at \(-5\) and an arrow pointing to the right (the first option in the original number - line choices) represents the solution set \(x\geq - 5\).